Determine whether the series is convergent or divergent. If it is convergent, find its sum.
Whether the series is convergent or divergent and obtain the sum if the series is convergent.
The series is convergent to the sum 2+2≈3.414.
The series is ∑k=0∞(2)−k.
The geometric series ∑n=1∞arn−1 (or) a+ar+ar2+⋯ is convergent if |r|<1 and its sum is a1−r, where a is the first term and r is the common ratio of the series.
The given series can be written as follows,
Here, the first term of the series is a = 1 and the common ratio of the series is,
The absolute value of r is,
Since |r|<1 and by using Result stated above, the series ∑k=0∞(2)−k is convergent.
Obtain the sum of the series.
Since a=1 and r=12.
Multiply by the conjugate 2+12+1 and simplify the terms,
Thus, the sum of the series is 2+2≈3.414.
Therefore, the series is convergent to the sum 2+2≈3.414.